Search arXiv⌕ Search

arXiv subjects

Vu Trung Hieu

Publications and source records attributed to Vu Trung Hieu.

13 recordsLinked to original sources
arXiv2026

A connectedness result for convex polynomial multi-objective optimization without boundedness

We prove that the weak Pareto solution set of a multi-objective optimization problem with convex polynomial objectives and constraints is connected, assuming neither boundedness of the constraint set nor a constraint qualification. Since this set is semi-algebraic, its connectedness implies that it is path-connected whenever it is nonempty. In particular, convex quadratic multi-objective optimization problems have connected weak Pareto solution sets; this answers the weak Pareto part of an open question posed by N.~D.~Yen in \emph{Recent Developments in Vector Optimization} (Springer, 2012).

math.OC↗
arXiv2025

Univariate representations of solutions to generic polynomial complementarity problems

By using the squared slack variables technique, we demonstrate that the solution set of a general polynomial complementarity problem is the image, under a specific projection, of the set of real zeroes of a system of polynomials. This paper points out that, generically, this polynomial system has finitely many complex zeroes. In such a case, we use symbolic computation techniques to compute a univariate representation of the solution set. Consequently, univariate representations of special solutions, such as least-norm and sparse solutions, are obtained. After that, enumerating solutions boils down to solving problems governed by univariate polynomials. We also provide some experiments on small-scale problems with worst-case scenarios. At the end of the paper, we propose a method for computing approximate solutions to copositive polynomial complementarity problems that may have infinitely many solutions.

math.OC↗
arXiv2024

Unboundedness of the images of set-valued mappings having closed graphs: Application to vector optimization

In this paper, we propose criteria for unboundedness of the images of set-valued mappings having closed graphs in Euclidean spaces. We focus on mappings whose domains are non-closed or whose values are connected. These criteria allow us to see structural properties of solutions in vector optimization, where solution sets can be considered as the images of solution mappings associated to specific scalarization methods. In particular, we prove that if the domain of a certain solution mapping is non-closed, then the weak Pareto solution set is unbounded. Furthermore, for a quasi-convex problem, we demonstrate two criteria to ensure that if the weak Pareto solution set is disconnected then each connected component is unbounded.

math.OC↗
arXiv2024

Computing local minimizers in polynomial optimization under genericity conditions

In this paper, we focus on computing local minimizers of a multivariate polynomial optimization problem under certain genericity conditions. By using a technique in computer algebra and the second-order optimality condition, we provide a univariate representation for the set of local minimizers. In particular, for the unconstrained problem, i.e. the constraint set is $\R^n$, the coordinates of all local minimizers can be represented by the values of $n$ univariate polynomials at real roots of a system including a univariate polynomial equation and a univariate polynomial matrix inequality. We also develop the technique for constrained problems having equality/inequality constraints. Based on the above technique, we design symbolic algorithms to enumerate the local minimizers and provide some experimental examples based on hybrid symbolic-numerical computations. For the case that the genericity conditions fail, at the end of the paper, we propose a perturbation technique to compute approximately a global minimizer provided that the constraint set is compact.

math.OC↗
arXiv2023

Disconnectedness and unboundedness of the solution sets of monotone vector variational inequalities

In this paper, we investigate the topological structure of solution sets of monotone vector variational inequalities. We show that if the weak Pareto solution set of a monotone vector variational inequality is disconnected, then each connected component of the set is unbounded. Similarly, this property holds for the proper Pareto solution set. Two open questions on the topological structure of the solution sets of (symmetric) monotone vector variational inequalities are raised at the end of the paper.

math.OC↗
arXiv2022

On the Nonemptiness and Boundedness of Solution Sets of Weakly Homogeneous Optimization Problems

In this paper, we introduce a new class of optimization problems whose objective functions are weakly homogeneous relative to the constraint sets. By using the normalization argument in asymptotic analysis, we prove two criteria for the nonemptiness and boundedness of the solution set of a weakly homogeneous optimization problem. Moreover, the normalization argument enables us to study the existence and stability of the solution sets of linearly parametric optimization problems. Several examples are given to illustrate the derived results.

math.OC↗
arXiv2021

On the solution existence and stability of polynomial optimization problems

This paper introduces and investigates a regularity condition in the asymptotic sense for optimization problems whose objective functions are polynomial. Under this regularity condition, the normalization argument in asymptotic analysis enables us to see the solution existence as well as the solution stability of these problems. We prove a Frank-Wolfe type theorem for regular optimization problems and an Eaves type theorem for non-regular pseudoconvex optimization problems. Moreover, we show results on the stability such as upper semicontinuity and local upper-Hölder stability of the solution map of polynomial optimization problems. At the end of the paper, we discuss the genericity of the regularity condition.

math.OC↗
arXiv2020

An Application of the Tarski-Seidenberg Theorem with Quantifiers to Vector Variational Inequalities

We study the connectedness structure of the proper Pareto solution sets, the Pareto solution sets, the weak Pareto solution sets of polynomial vector variational inequalities, as well as the connectedness structure of the efficient solution sets and the weakly efficient solution sets of polynomial vector optimization problems. By using the Tarski-Seidenberg Theorem with quantifiers, we are able to prove that these solution sets are semi-algebraic without imposing the Mangasarian-Fromovitz constraint qualification on the system of constraints.

math.OC↗
arXiv2020

Solution maps of polynomial variational inequalities

In this paper, we investigate several properties of the solution maps of variational inequalities with polynomial data. First, we prove some facts on the $R_0$-property, the local boundedness, and the upper semicontinuity of the solution maps. Second, we establish results on the solution existence and the local upper-Hölder stability under the copositivity condition. Third, when the constraint set is semi-algebraic, we discuss the genericity of the $R_0$-property and the finite-valuedness of the solution maps.

math.OC↗
arXiv2020

Dimension in Polynomial Variational Inequalities

The aim of the paper is twofold. Firstly, by using the constant rank level set theorem from differential geometry, we establish sharp upper bounds for the dimensions of the solution sets of polynomial variational inequalities under mild conditions. Secondly, a classification of polynomial variational inequalities based on dimensions of their solution sets is introduced and investigated. Several illustrative examples are provided.

math.OC↗
arXiv2018

On the R$_0$-tensors and the solution map of tensor complementarity problems

Our purpose is to investigate the local boundedness, the upper semicontinuity, and the stability of the solution map of tensor complementarity problems. To do this, we focus on the set of R$_0$--tensors and show that this set plays an important role in the investigation. Furthermore, by using a technique in semi-algebraic geometry, we obtain some results on the finite-valuedness and the lower semicontinuity of the solution map.

math.OC↗
arXiv2018

Numbers of the connected components of the solution sets of monotone affine vector variational inequalities

This paper establishes several upper and lower estimates for the maximal number of the connected components of the solution sets of monotone affine vector variational inequalities. Our results give a partial solution to Question~2 in [N.D. Yen and J.-C. Yao, \textit{Monotone affine vector variational inequalities}, Optimization 60 (2011), pp. 53--68] and point out that the number depends not only on the number of the criteria but also on the number of variables of the vector variational inequality under investigation.

math.OC↗